Now, I understand that dividing by zero in any case is undefined. However, in math, there are always exceptions. I'm just really curious...what are the different cases for different answers? For most controversial arguments, I've looked up both sides, but I can't seem to find that many sources for this one.
Personally, I think it $\frac 00$ is 0 because for younger kids, division can often be a word problem such as the following:
If John has 20 apples, and he divides his apples up between his 5 friends, how many apples did he give each of his friends?
Obviously, the answer is 4 apples. Now, reword the problem to say:
If John has 0 apples, and he divides his apples up between his 5 friends, how many apples did he give each of his friends?
Now the answer, given this context, is 0. Of course, once the decimal system is introduced in a person's education, and the "remainder" system is useless (eg, $\frac54 = 1 $, R $ 1$), this proof no longer works.
TL;DR: If there is nothing there, and you don't divide it, there's still nothing there. You can't magically create 1 to satisfy the $\frac aa = 1$ rule.
I'm trying really hard to figure out exactly what you are asking, but here's my best answer.
For any real $c$,
Case 1: denominator approaches zero from the right $$\lim_{x\to 0^+}\frac{c}{x}=\infty$$
Case 2: denominator approaches zero from the left $$\lim_{x\to 0^-}\frac{c}{x}=-\infty$$
Case 3: numerator approaches zero from both directions $$\lim_{x\to 0}\frac{x}{c}=0$$
Case 4: numerator and denominator approach zero from both directions $$\lim_{x\to 0}\frac{x}{x}=1$$
Maybe you could rephrase your question to be more specific. In the mean time, do some light reading so you can refine your question.